Question

If tan θ + sin θ = m, tan θ– sin θ = n then show that m² – n² = 4 √mn

Answer 1

SOLUTION IS IN THE ATTACHMENT

Trigonometry is the study of the relationship between the sides and angles of a triangle.

An equation involving trigonometry ratios of an angle is called is called a trigonometric  identity, if it is true for all values of the angles involved. For any acute angle θ, we have 3 identities.

i) sin² θ + cos² θ = 1 ,ii) 1 + tan² θ = sec² θ , iii) cot² θ +1 = cosec² θ.

HOPE THIS WILL HELP YOU...

Answer 2

Hello .


tan∅ + Sin∅ = m

tan∅ - Sin∅ = n


m² = ( tan∅ + sin∅ ) ²
= tan²∅ + sin²∅ + 2tan∅ +sin∅

n² = ( tan∅ - sin∅ ) ²
= tan²∅ + sin²∅ - 2 tan∅sin∅

Now , m²-n²

= tan²∅ + sin²∅ + 2tan∅sin∅ - tan²∅ - sin²∅ + 2tan∅sin0
= 4tan∅sin∅ .
= 4sin∅ / cos∅ * sin∅
= 4sin²∅ / cos∅

Now ,

mn = ( tan∅ + sin∅ ) ( tan∅ - sin∅ )
= tan²∅ - sin²∅
= sin²∅ / cos²∅ - sin²∅
= sin²∅ - sin²∅cos²∅ / cos²∅
= sin²∅ ( 1 - cos²0 ) / cos²∅
= sin²∅ * sin²∅ / cos²∅
= sin⁴∅ / cos²∅

√mn = √ sin⁴∅/ cos²∅
√mn = sin²∅ / cos∅ .

4√mn = 4* sin²∅ / cos∅

So ,

m²-n² = 4mn

kul !!
Regards - Me